Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms

Chunhua  Wang; Jing  Yang

doi:10.3934/dcds.2016.36.1603

Article Contents

2016, Volume 36, Issue 3: 1603-1628. Doi: 10.3934/dcds.2016.36.1603

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Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms

Chunhua Wang^1, and
Jing Yang^1,

1.
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079

Received: November 2014

Revised: April 2015

Published: May 2017

Abstract / Introduction Related Papers Cited by

Abstract

In this paper, by an approximating argument, we obtain infinitely many solutions for the following problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta u = \mu \frac{|u|^{2^{*}(t)-2}u}{|y|^{t}} + \frac{|u|^{2^{*}(s)-2}u}{|y|^{s}} + a(x) u, & \hbox{$\text{in} \Omega$}, \\ u=0,\,\, &\hbox{$\text{on}~\partial \Omega$}, \\ \end{array} \right. \end{equation*} where $\mu\geq0,2^{*}(t)=\frac{2(N-t)}{N-2},2^{*}(s) = \frac{2(N-s)}{N-2},0\leq t < s < 2,x = (y,z)\in \mathbb{R}^{k} \times \mathbb{R}^{N-k},2 \leq k < N,(0,z^*)\subset \bar{\Omega}$ and $\Omega$ is an open bounded domain in $\mathbb{R}^{N}.$ We prove that if $N > 6+t$ when $\mu>0$ and $N > 6+s$ when $\mu=0,$ $a((0,z^*)) > 0,$ $\Omega$ satisfies some geometric conditions, then the above problem has infinitely many solutions.

Keywords:

Double critical Hardy-Sobolev-Maz'ya terms,
infinitely many solutions,
variational methods.

Mathematics Subject Classification: 35J60, 35B33.

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